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*Deep Down Things: The Breathtaking Beauty of Particle Physics*by Bruce A. Schumm. He explains the Time-independent Schrödinger equation as a statement of the conservation of energy. You put in the function V(x) to describe the potential energy of the system, and then you can solve for ψ(x). I follow all that.

Later, he discusses local gauge invariance, and added a term A(x) which allows you to exactly undo any arbitrary phase changes you made in ψ; then goes on to identify A with an electric field and points out that the phase doesn't matter, so introducing local gauge invariance rather than being an ad-hoc fix inspired a way to handle all the bosons in the Lie group (1 in the case of em-field).

So, the function A represents a force field.

If you put the things influencing the particle in the A function (nearby electric charges), then what is the V function used for? Wasn't that being used to represent the potential energy of being near other charged objects? What am I missing?

TIA,

—John